Recall that the choice of the random set (REF ) in the P-step of Section REF  was justified by the structure of the function \(\psi _i\)  as a non-conformity measure. There, we considered mappings \(\psi _i\)  such that smaller values of them are “better” in the sense that \(y_i\)  agrees with the prediction derived based on data \(y_{-i}^{n+1}\) . However, this is not necessarily the case for \(\psi _i\)  in (REF ), as we are merely returning \(y_i\)  when comparing it to \(y_{-i}^{n+1}\) . How to motivate a particular choice of random set \(\mathcal {S}\) ? [1]} consider different classes of assertions/hypotheses about \(Y_{n+1}\) , and the structure of these assertions can inform the choice the random set. For one-sided assertions, e.g., \(\lbrace Y_{n+1} \le \tilde{y}\rbrace \)  or \(\lbrace Y_{n+1} \ge \tilde{y}\rbrace \) , for varying \(\tilde{y}\) , the natural choice of random set is similarly one-sided.
Here, however, we will focus on singleton assertions, \(\lbrace Y_{n+1} = \tilde{y}\rbrace \) , and the corresponding IM will produce two-sided prediction intervals. Specifically, for the P-step, we recommend
\(\mathcal {S}= \bigl \lbrace \tfrac{n+2}{2} - |\tilde{V} - \tfrac{n+2}{2}|,\ldots ,\tfrac{n+2}{2} + |\tilde{V} - \tfrac{n+2}{2}|\bigr \rbrace , \quad \tilde{V} \sim {\sf Unif}(I_{n+1}),\) 
